Embedding Planar Graphs at Fixed Vertex Locations

 Let G be a planar graph of n vertices, v ₁,…,v _n , and let {p ₁,…,p _n } be a set of n points in the plane. We present an algorithm for constructing in O(n ²) time a planar embedding of G, where vertex v _i is represented by point p _i and each edge is represented by a polygonal curve with O(n) bends (internal vertices). This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of G mapping vertices to their assigned points and edges to polygonal curves has at least m/20 edges represented by curves with at least m/40³ bends. Received: May 24, 1999 Final version received: April 10, 2000     

Near-proper vertex 2-colorings of sparse graphs

A graph G is (2, 1)-colorable if its vertices can be partitioned into subsets V ₁ and V ₂ such that each component in G[V ₁] contains at most two vertices while G[V ₂] is edgeless. We prove that every graph with maximum average degree mad(G) < 7/3 is (2, 1)-colorable. It follows that every planar graph with girth at least 14 is (2, 1)-colorable. We also construct a planar graph G _n with mad (G _n ) = (18n − 2)/(7n − 1) that is not (2, 1)-colorable.     

A Degree Constraint for Uniquely Hamiltonian Graphs

A graph, G, is called uniquely Hamiltonian if it contains exactly one Hamilton cycle. We show that if G=(V, E) is uniquely Hamiltonian then Where #(G)=1 if G has even number of vertices and 2 if G has odd number of vertices. It follows that every n-vertex uniquely Hamiltonian graph contains a vertex whose degree is at most c log₂n+2 where c=(log₂3−1)⁻¹≈1.71 thereby improving a bound given by Bondy and Jackson [3].     

Metric Dimension and R-Sets of Connected Graphs

The R-set relative to a pair of distinct vertices of a connected graph G is the set of vertices whose distances to these vertices are distinct. This paper deduces some properties of R-sets of connected graphs. It is shown that for a connected graph G of order n and diameter 2 the number of R-sets equal to V(G) is bounded above by ?n²/4?{\lfloor n^{2}/4\rfloor} . It is conjectured that this bound holds for every connected graph of order n. A lower bound for the metric dimension dim(G) of G is proposed in terms of a family of R-sets of G having the property that every subfamily containing at least r ≥ 2 members has an empty intersection. Three sufficient conditions, which guarantee that a family F=(G_n)_{n 3 1}{\mathcal{F}=(G_{n})_{n\geq 1}} of graphs with unbounded order has unbounded metric dimension, are also proposed.     

Total Domination in Graphs with Given Girth

A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ _t (G) of G. In this paper, we establish an upper bound on the total domination number of a graph with minimum degree at least two in terms of its order and girth. We prove that if G is a graph of order n with minimum degree at least two and girth g, then γ _t (G) ≤ n/2 + n/g, and this bound is sharp. Our proof is an interplay between graph theory and transversals in hypergraphs. Michael A. Henning: Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Open Rectangle-of-Influence Drawings of Inner Triangulated Plane Graphs

A straight-line drawing of a plane graph is called an open rectangle-of-influence drawing if there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of every edge. In an inner triangulated plane graph, every inner face is a triangle although the outer face is not necessarily a triangle. In this paper, we first obtain a sufficient condition for an inner triangulated plane graph G to have an open rectangle-of-influence drawing; the condition is expressed in terms of a labeling of angles of a subgraph of G. We then present an O(n ^1.5/log n)-time algorithm to examine whether G satisfies the condition and, if so, construct an open rectangle-of-influence drawing of G on an (n−1)×(n−1) integer grid, where n is the number of vertices in G.     

On the total domination subdivision numbers in graphs

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_γt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sd_{γ t} (G) ≤ 2γ _t (G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sd_{γ t} (G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sd_{γ t} (G)=2γ _t (G)−1.     

Extension of Turán's Theorem to the 2-Stability Number

 Given a graph G with n vertices and stability number α(G), Turán's Theorem gives a lower bound on the number of edges in G. Furthermore, Turán has proved that the lower bound is only attained if G is the union of α(G) disjoint balanced cliques. We prove a similar result for the 2-stability number α₂(G) of G, which is defined as the largest number of vertices in a 2-colorable subgraph of G. Given a graph G with n vertices and 2-stability number α₂(G), we give a lower bound on the number of edges in G and characterize the graphs for which this bound is attained. These graphs are the union of isolated vertices and disjoint balanced cliques. We then derive lower bounds on the 2-stability number, and finally discuss the extension of Turán's Theorem to the q-stability number, for q>2. Received: July 21, 1999 Final version received: August 22, 2000 Present address: GERAD, 3000 ch. de la Cote-Ste-Catherine, Montreal, Quebec H3T 2A7, Canada. e-mail: Alain.Hertz@gerad.ca     

List‐coloring the square of a subcubic graph

The square G² of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G²) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list‐chromatic number of G² equals the chromatic number of G², that is, χ_l(G²) = χ(G²) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χ_l(G²) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χ_l(G²) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χ_l(G²) ≤ 7. Dvo?ák, ?krekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χ_l(G²) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χ_l(G²) ≤ 6. All of our proofs can be easily translated into linear‐time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008     

On the chromatic polynomial of a graph

Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()^α≤≤ () ^{n −ω}. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour number μ(G) of G: n− (n−ω)() ^{n −ω}≤μ(G)≤n−α() ^α. Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

On Cycles in 3-Connected Graphs

 Let G be a 3-connected graph of order n and S a subset of vertices. Denote by δ(S) the minimum degree (in G) of vertices of S. Then we prove that the circumference of G is at least min{|S|, 2δ(S)} if the degree sum of any four independent vertices of S is at least n+6. A cycle C is called S-maximum if there is no cycle C ^′ with |C ^′∩S|>|C∩S|. We also show that if ∑⁴ _i=1 d(a _i)≥n+3+|⋂⁴ _i=1 N(a _i)| for any four independent vertices a ₁, a ₂, a ₃, a ₄ in S, then G has an S-weak-dominating S-maximum cycle C, i.e. an S-maximum cycle such that every component in G−C contains at most one vertex in S. Received: March 9, 1998 Revised: January 7, 1999     

On total chromatic number of planar graphs without 4-cycles

Let G be a simple graph with maximum degree A(G) and total chromatic number Xve(G). Vizing conjectured thatΔ(G) 1≤Xve(G)≤Δ(G) 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs isΔ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then Xve(G)≤8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.     

Gromov hyperbolic cubic graphs

If X is a geodesic metric space and x ₁; x ₂; x ₃ ∈ X, a geodesic triangle T = {x ₁; x ₂; x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.     

Hamiltonian Cycles in Almost Claw-Free Graphs

 Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G ₁, G ₂ and G ₃ such that E _G (G _i , G _j ) = {u _i , u _j , v _i v _j } for i≠j and i,j = 1, 2, 3 (where u _i ≠v _i ∈V(G _i ) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free graphs. Received: September 21, 1998 Final version received: August 18, 1999     

Injective (Δ + 1)-coloring of planar graphs with girth 6

A vertex coloring of a graph G is called injective if every two vertices joined by a path of length 2 get different colors. The minimum number χ _i (G) of the colors required for an injective coloring of a graph G is clearly not less than the maximum degree Δ(G) of G. There exist planar graphs with girth g ≥ 6 and χ _i = Δ+1 for any Δ ≥ 2. We prove that every planar graph with Δ ≥ 18 and g ≥ 6 has χ _i ≤ Δ + 1.     

Tiling Transitive Tournaments and Their Blow-ups

Let TT _k denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT _k by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c ₂=1/2, c ₃=5/6 and c _k =1−2^{−k−log k} for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c _k n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c _k cannot be improved to less than 1−2^{−0.5k(1+o(1))}. This revised version was published online in June 2006 with corrections to the Cover Date.     

Set intersection representations for almost all graphs

Two variations of set intersection representation are investigated and upper and lower bounds on the minimum number of labels with which a graph may be represented are found that hold for almost all graphs. Specifically, if θ_k(G) is defined to be the minimum number of labels with which G may be represented using the rule that two vertices are adjacent if and only if they share at least k labels, there exist positive constants c_k and c′_k such that almost every graph G on n vertices satisfies Changing the representation only slightly by defining θ;_odd (G) to be the minimum number of labels with which G can be represented using the rule that two vertices are adjacent if and only if they share an odd number of labels results in quite different behavior. Namely, almost every graph G satisfies Furthermore, the upper bound on θ_odd(G) holds for every graph. © 1996 John Wiley & Sons, Inc.     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

A note on constructive methods for ramsey numbers

Let r(k) denote the least integer n-such that for any graph G on n vertices either G or its complement G contains a complete graph K_k on k vertices. in this paper, we prove the following lower bound for the Ramsey number r(k) by explicit construction: r(k) ≥ exp (c(Log k)^4/3[(log log k)^1/3] for some constant c> 0.